The cloud office is your day-to-day in the cloud. It's a new platform for email and documents and messaging and calendaring, but more than that, it's connectivity; it's collaboration; it's centralization; it's having choices about where and when you work.
Microsoft Office 365 is a subscription-based service that's accessed through the Internet on a web browser. It's not stored on your computer—You must launch it through a web browser each time you use it. SaaS applications can be run in the Cloud, but this doesn't make them a Cloud.